Is it possible to show the existence of harmonic coordinates (e.g., on uniformsized balls) on certain classes of noncompact Riemannian manifolds? For example, one may expect that such harmonic coordinates exist on a manifold that has doubling volume measure and supports Poincare inequality?

3$\begingroup$ A key word is harmonic radius, see these notes www1.mat.uniroma1.it/ricerca/rendiconti/ARCHIVIO/1997(4)/… $\endgroup$– Andy SandersCommented Feb 12, 2019 at 5:02

$\begingroup$ Thank you! Indeed, I want to ask the following: can we get a uniform lower bound on the harmonic radius, provided that the noncompact manifold has doubling volume measure and supports Poincare inequality? $\endgroup$– Siran Victor LiCommented Feb 12, 2019 at 22:38

$\begingroup$ Thank you Andy! I think the corollary on p.577 or HebeyHerzlich answers my question. $\endgroup$– Siran Victor LiCommented Feb 13, 2019 at 1:52
1 Answer
The Main Lemma 2.2 in "Michael T. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds" (http://link.springer.com/article/10.1007/BF01233434) says essentially that there is a uniform lower bound for the harmonic radius in term of Ricci curvature bound and lower bound of injectivity radius.
The paper "E. HEBEY & M. HERZLICH, Harmonic coordinates, harmonic radius and convergence of Riemannian manifolds" (http://www1.mat.uniroma1.it/ricerca/rendiconti/ARCHIVIO/1997(4)/569605.pdf) is an exposition of the above paper of Anderson, which is, in my viewpoint, easier to read.

$\begingroup$ Thank you very much Truong! I think the corollary on p.577 or HebeyHerzlich answers my question. $\endgroup$ Commented Feb 13, 2019 at 1:52